MANCHESTER, N.H. — Bernie Sanders is 74. He grew up playing stickball in the streets of Brooklyn and watching a black-and-white television.

Yet this child of the 1940s, who says Franklin D. Roosevelt is his favorite president, has inspired a potent political movement among young people today. College students wear shaggy white “Bernie” wigs on campus, carry iPhones with his image as their screen saver and flock to his events by the thousands.

And armies of young voters are turning what seemed like a long-shot presidential candidacy into a surprisingly competitive campaign.

“He may seem like some old geezer who doesn’t care about stuff,” said Caroline Buddin, 24, a sales associate in Charleston, S.C. “But if you actually give him the time of day and listen to what he has to say, he has a lot of good ideas.”

In interviews, young supporters of the Vermont senator’s presidential bid almost all offer some version of the same response when asked why they like him: He seems sincere.

For the generation that researchers say has been the most bombarded with marketing slogans and advertising pitches, Mr. Sanders, a former mayor of Burlington, Vt., has a certain unpolished appeal.

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Some have taken to wearing shaggy white “Bernie” wigs to show their support.CreditTodd Heisler/The New York Times

The first group of students working to elect Mr. Sanders president sprang up at Middlebury College in Vermont. There are now similar chapters at more than 220 campuses across the country, with the biggest one at the University of California at Berkeley.

The movement, at least initially, was not so much the result of an organized effort by the Sanders campaign, but more of a visceral response to the candidate himself

“It seems like he is at the point in his life when he is really saying what he is thinking,” said Olivia Sauer, 18, a college freshman who returned to her hometown, Ames, Iowa, to caucus for Mr. Sanders.

“With Hillary,” she said, “sometimes you get this feeling that all of her sentences are owned by someone.”

Young voters’ support for Mr. Sanders has created a quandary in Hillary Clinton’s campaign headquarters in Brooklyn, where millennial staff members have tried to persuade their peers to back the former secretary of state, using social media platforms like Snapchat and Instagram. On Monday in Iowa, Mr. Sanders defeated Mrs. Clinton among voters ages 17 to 29 by 70 percentage points, greater than the 43-percentage-point margin by which Barack Obama won that age group in Iowa in 2008.

That is true among men and women, and even Mrs. Clinton called the gap “amazing” during an appearance on CNN on Wednesday.

The day after Mrs. Clinton, 68, eked out a win over Mr. Sanders in Iowa, her campaign held a conference call with prominent Democratic supporters. Several of the officials, who are delegates to the party’s convention, expressed concerns about the campaign’s sometimes awkward attempts to reach young voters, including its reliance on baby boomer celebrities who have less resonance with the millennial generation.

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Hillary Clinton’s campaign says she will win over young voters when they learn more about her proposals to make college affordable and combat climate change.CreditRichard Perry/The New York Times

“I’m very close to Jamie Lee Curtis and I know she was there, but she’s not young anymore,” Rosalind Wyman, a prominent Democrat in Los Angeles, said of the 57-year-old actress and spokeswoman for Activia yogurt, who campaigned for Mrs. Clinton in Iowa.

On the phone call, Marlon Marshall, the director of state campaigns and political engagement, tried to assure the group that the Clinton operation would be using more youthful surrogates, including Mrs. Clinton’s 35-year-old daughter, Chelsea, and would continue to talk about issues like college affordability.

“I do believe a narrative will continue,” Mr. Marshall said. The Sanders campaign, he added, will “keep drumming that.”

Mrs. Clinton’s support among young voters has fallen in the eight years since she last ran for president. In the 2008 Democratic primary in New Hampshire, 60 percent of voters ages 18 to 24 supported Mr. Obama, while those ages 25 to 29 split their support between the two candidates, according to exit polls conducted by Edison Research.

Some 87 percent of likely New Hampshire primary voters ages 18 to 29 said they would vote for Mr. Sanders in the state’s primary on Tuesday, compared with 13 percent for Mrs. Clinton, according to a UMass-Lowell poll conducted Feb. 1 to 3.

Sarah King-Mayes, 18, a freshman at the University of New Hampshire, volunteers three or four times a week for the Sanders campaign. She says she plans to register to vote on Tuesday in New Hampshire, instead of waiting to vote in Massachusetts, where her family lives.

She started supporting Mr. Sanders last summer after she took an online quiz, which concluded her views most matched those of the senator, particularly on issues like racial injustice and climate change.

“I am excited for the political revolution,” Ms. King-Mayes said. “Because I am excited for an actual democracy to be reinstated where politicians cannot be bought out by whatever company, I guess, wants their votes.”

Mrs. Clinton’s campaign says she will win over young voters when they learn more about her proposals to make college affordable and to combat climate change. “The difference for young voters and for all voters is the ability to get something done,” Robby Mook, her campaign manager said in an interview with MSNBC this week. “And what that takes is someone in Washington who can break through the gridlock.”

That argument won over Iris Brenner, 21, a student at Iowa State University in Ames, who supports Mrs. Clinton. “For me, it’s been, ‘Do I caucus for someone who is a little less exciting but who can get stuff done?’”

Privately, Mrs. Clinton’s supporters say that while being a youth icon has its advantages, the support of middle-aged and older voters is enough for her to capture the nomination. In the Iowa caucuses, she beat Mr. Sanders by 23 points among voters ages 45 to 64 and by 43 points among voters 65 and older, according to exit polls.

“They don’t have to be for me, I’m going to be for them,” Mrs. Clinton told CNN on Wednesday when asked about the generation gap.

The Sanders campaign said excitement among young people had been organic, although it has clearly worked to improve the candidate’s appeal on sites like Twitter, Reddit and Snapchat, and in text messages to supporters. “Young people are idealistic and they look at this country and say we can be much more,” Mr. Sanders told reporters Wednesday.

The discomfort, and, in some cases hostility toward Mrs. Clinton among young voters is striking. Some of them, feeling the pinch of economic hardship or the burden of college loans, suggest she is too cozy with big banks and corporate America. But polls also show they do not find her trustworthy.

The scene at the party for Mr. Sanders on the night of the Iowa caucuses captured their feelings. As Mrs. Clinton appeared on several large projector screens, telling her supporters she was “breathing a sigh of relief” to learn of the Iowa results, some young members of the Sanders crowd booed loudly.

“I am a progressive who gets things done for people,” Mrs. Clinton said, setting off more heckling from the Sanders supporters. “She’s a liar,” the group chanted angrily.

Even those showing up to the Clinton campaign’s youth-oriented events do not seem to be embracing her candidacy.

Last month in Iowa, Mrs. Clinton appeared at a joint rally in Iowa City with the 23-year-old pop star Demi Lovato. Like many college students in the crowd, Nate Weger, 20, said he was supporting Mr. Sanders but was not going to say no to a free Demi Lovato concert.

“Bernie just seems a lot more honest when he talks,” Mr. Weger said. Similar sentiments were voiced at an Iowa event for Mrs. Clinton featuring Lena Dunham, the creator and star of the HBO show “Girls.”

Pete D’Alessandro, Mr. Sanders’s campaign director in Iowa, said such efforts could be futile. He recalled a story from 1968, during Robert F. Kennedy’s primary fight against Eugene McCarthy, when the McCarthy campaign held a barbecue that attracted hundreds of voters, leading some to believe he had deep support.